A family of random walks with generalized Dirichlet steps

TL;DR

This paper studies a class of continuous-time random walks in multi-dimensional space with steps following a generalized Dirichlet distribution, deriving explicit probability densities and extending previous Dirichlet-based models.

Contribution

It provides explicit formulas for the probability density functions of these generalized Dirichlet random walks, including special cases and unconditional distributions using fractional Poisson processes.

Findings

Explicit density functions for the position process are derived.

Special cases with two steps are analyzed in detail.

Unconditional distributions are obtained via fractional Poisson processes.

Abstract

We analyze a class of continuous time random walks in $\mathbb R^d,d\geq 2,$ with uniformly distributed directions. The steps performed by these processes are distributed according to a generalized Dirichlet law. Given the number of changes of orientation, we provide the analytic form of the probability density function of the position $\{\underline{\bf X}_d(t),t>0\}$ reached, at time $t>0$, by the random motion. In particular, we analyze the case of random walks with two steps. In general, it is an hard task to obtain the explicit probability distributions for the process $\{\underline{\bf X}_d(t),t>0\}$ . Nevertheless, for suitable values for the basic parameters of the generalized Dirichlet probability distribution, we are able to derive the explicit conditional density functions of $\{\underline{\bf X}_d(t),t>0\}$. Furthermore, in some cases, by exploiting the fractional Poisson process, the unconditional probability distributions are obtained. This paper extends in a more general setting, the random walks with Dirichlet displacements introduced in some previous papers.